How much can one learn from a single solution of a PDE?
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Linear evolution PDE ∂_t u(x,t) = -ℒ u, where ℒ is a strongly elliptic operator independent of time, is studied as an example to show if one can superpose snapshots of a single (or a finite number of) solution(s) to construct an arbitrary solution. Our study shows that it depends on the growth rate of the eigenvalues, μ_n, of ℒ in terms of n. When the statement is true, a simple data-driven approach for model reduction and approximation of an arbitrary solution of a PDE without knowing the underlying PDE is designed. Numerical experiments are presented to corroborate our analysis.
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